Is inertial mass of gas the same as its rest mass

Suppose we have a box at rest that is filled with a uniform gas. We denote the volume by V and the pressure by p. Suppose next that we apply a small force to the box and accelerate it until it has a speed v. The key question is: Is it harder to accelerate the gas because it takes work not only to accelerate the existing energy but also to compress the gas as the Lorentz contraction demands? In other words, is the inertia of the box is larger than just its rest mass?